Regarding supremum over a set

Let $$\Delta$$ be a Compact Hausdorff space in $$\mathbb{C^n}$$. Let $$A$$ be a closed sub algebra of $$C(\Delta)$$(space of all complex valued continuous functions on $$\Delta$$) which contains the constant functions. Let $$\mathbb{D}$$ be the open unit ball in the complex plane.

Let $$A{(\mathbb{D})}=\{f\in C(\Delta): f(\Delta)\subset\mathbb{D} \}$$ and $$A({\bar{\mathbb{D}}})=\{f\in C(\Delta): f(\Delta)\subset\bar{\mathbb{D}} \}$$.

Now let $$a,b\in\Delta$$, and consider the supremum $$\sup_f\left\{\left|\frac{f(w_1)-f(w_2)}{1-\overline{f(w_2)}f(w_1)}\right|: f\in A{(\mathbb{D})} \right\}$$. It is said that the above defined supremum is the same if we take $$A({\bar{\mathbb{D}}})$$ or $$A{(\mathbb{D})}$$. Can anyone tell how?

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• "It is said" where or by whom? – gmvh Oct 15 at 14:53
• Line under equation (6) on page 812 of this article worldscientific.com/doi/10.1142/S0129167X97000408 – Ma18 Oct 15 at 15:30
• Given $f\in A(\overline{\mathbb D})$ you have $rf\in A(\mathbb D)$ for every $r<1$. Then take limits $r\to 1$. – Jochen Wengenroth Oct 16 at 11:08